Washington -LRB- CNN -RRB- Former Florida Gov. Jeb Bush 's surprise announcement that he 'll `` actively explore '' a presidential bid Tuesday morning did little to dissuade his potential Republican opponents from the race .

Following the news , South Carolina Sen. Lindsey Graham , who might make a White House bid , offered the clearest pitch yet for his own candidacy .

`` I think I have a unique ability to deal with the threats we face at home and abroad and the challenges here , which is finally getting the government to work and dealing with a dangerous world , '' he told reporters on Capitol Hill .

He said to `` stay tuned '' for his plans .

`` I think there are a lot of people in the donor class who are looking for multiple voices , including Jeb 's , and competition is a good thing , '' Graham said , adding , `` He 's got a lot to offer the Republican Party and the country . ''

Kentucky Sen. Rand Paul was upbeat about the governor 's announcement .

`` I think we 're a big tent -- we can use moderates , conservatives , libertarians -- we need 'em all , '' he told reporters . `` I think the more the merrier -- the public will determine '' whether Bush could win .

And Florida Sen. Marco Rubio 's spokesman , Alex Conant , said while Rubio has `` a lot of respect '' for Bush and thinks he 'd make `` a formidable candidate , '' Bush 's decision has no bearing on his own .

`` Marco 's decision on whether to run for president or re-election will be based on where he can best achieve his agenda to restore the American Dream -- not on who else might be running , '' Conant said .

Many observers believe if Bush jumps in the race , Rubio -- who 's been mentored by the former Florida governor since his entry into politics -- would n't run . But a source close to Rubio said Tuesday that while Tuesday 's announcement changes the dynamic of the race , the junior Florida senator is still `` on track to run for president '' in 2016 .

Another potential opponent , Texas Sen. Ted Cruz , barely acknowledged that the two could face off in the primary fight .

`` I think what Republicans want is someone who will stand up and lead , who will take on the great challenges of the day , who will make the case that the path we 're on is n't working ; that the Obama-economy is a disaster , that Obamacare is a trainwreck , that our constitutional rights are under assault and that we need to restore America 's leadership in the world , '' he said .

Sen. John Thune of South Dakota , who recently left the door open to his own 2016 bid , said the Bush announcement was `` kind of expected . '' Now , though , `` everyone is starting to feel pressure to start making moves , '' he said .

But even though Bush is seen as one of the top establishment picks in the primary if he runs , it was clear Tuesday morning he 'll still have some work to convince some party leaders .

Arizona Sen. John McCain , the GOP 's 2008 presidential nominee , would n't disclose what he discussed in his recent meeting with Bush , only that they went over the `` ins and outs '' of a presidential campaign .

He would n't back the governor , either , since he 's still waiting to see whether Graham , who McCain called his `` closest friend , '' decides to run .

Bush would enter the race with his `` contributors , his family name , a long record as very successful governor , '' McCain said .

Sen. Rob Portman , considered a possible vice presidential pick , told reporters he has talked to Bush recently about a potential campaign .

`` He 'll be a strong candidate , '' said Portman , who recently announced he would be running for re-election rather than the White House in 2016 .

None of the GOP senators noted one of Bush 's biggest liabilities in a GOP presidential primary , however -- his support for policies anathema to conservative Republicans . Bush 's support for Common Core educational standards and his previous openness to a pathway to citizenship have drawn him many critics on the far-right , and he 'll face a challenge in winning over some of those primary voters .

Soon after his announcement , the Conservative Action Fund , an outside spending group , emailed its supporters asking them to sign a petition urging Bush not to run and decrying him as `` another establishment , compromising '' Republican .

But even right-wing firebrand Cruz on Tuesday held his fire when given the opportunity to attack Bush on one of his primary vulnerabilities , immigration reform .

`` Immigration will be a critical issue in this next election because we have a President who has been unwilling to work with Congress on common sense immigration reform , '' Cruz said , when asked how Bush 's position on immigration would play out in the primaries .

And some Republicans on Capitol Hill argued Bush still had a lot of work to do simply building the operation and support he 'd need to compete in a primary .

New Hampshire Sen. Kelly Ayotte , also seen as a possible vice presidential pick , noted Bush has yet to stop by her state -- which hosts the first primary in the nation -- to meet with grassroots activists .

`` I do n't care who you are , you have to do the hard work in New Hampshire , '' she said . `` You got ta do the grassroots . You got ta get down there and talk to the activists . ''

She acknowledged that he 'd have `` the Bush name recognition , '' but said there 's no substitute for hard retail politicking in the Granite State .

`` It will make a difference how hard the candidates work in terms of meeting the activists and answering the hard questions , '' Ayotte said .

CNN 's Ted Barrett contributed to this report

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Sen. Rand Paul welcomed Bush into the race : `` The more the merrier ! ''

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A spokesman for Sen. Marco Rubio said Bush 's decision would have no impact on his own

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Sen. Ted Cruz said the voters would decide on their eventual nominee